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Global solutions for the gravity water waves system in 2d

Alexandru D. Ionescu, Fabio Pusateri

TL;DR

The paper proves global existence and detailed long-time behavior for the gravity water waves system with a 1D interface in 2D, for small, smooth, localized data. It combines Wu’s Lagrangian framework with an Eulerian analysis, introducing a robust Z′-norm to capture decay and a nonlinear phase correction for modified scattering. The strategy hinges on a two-pronged approach: transforming quadratic nonlinearities to cubic via a carefully constructed normal form and controlling the resulting energies in Wu’s good coordinates, while simultaneously tracking dispersive decay through Eulerian variables. The work yields the first global, smooth, nontrivial 2D gravity water waves solutions and reveals a distinct asymptotic regime from the linear case, marked by a nonlinear logarithmic correction.

Abstract

We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu \cite{WuAG}. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case \cite{GMS2,Wu3DWW}. In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2d.

Global solutions for the gravity water waves system in 2d

TL;DR

The paper proves global existence and detailed long-time behavior for the gravity water waves system with a 1D interface in 2D, for small, smooth, localized data. It combines Wu’s Lagrangian framework with an Eulerian analysis, introducing a robust Z′-norm to capture decay and a nonlinear phase correction for modified scattering. The strategy hinges on a two-pronged approach: transforming quadratic nonlinearities to cubic via a carefully constructed normal form and controlling the resulting energies in Wu’s good coordinates, while simultaneously tracking dispersive decay through Eulerian variables. The work yields the first global, smooth, nontrivial 2D gravity water waves solutions and reveals a distinct asymptotic regime from the linear case, marked by a nonlinear logarithmic correction.

Abstract

We consider the gravity water waves system in the case of a one dimensional interface, for sufficiently smooth and localized initial data, and prove global existence of small solutions. This improves the almost global existence result of Wu \cite{WuAG}. We also prove that the asymptotic behavior of solutions as time goes to infinity is different from linear, unlike the three dimensional case \cite{GMS2,Wu3DWW}. In particular, we identify a suitable nonlinear logarithmic correction and show modified scattering. The solutions we construct in this paper appear to be the first global smooth nontrivial solutions of the gravity water waves system in 2d.

Paper Structure

This paper contains 110 sections, 54 theorems, 803 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. The main theorem
  4. Main ideas in the proof
  5. Supercriticality, energy estimates and normal forms
  6. Dispersion and asymptotic behavior
  7. Plan of the paper
  8. Strategy of the proof
  9. Local Existence
  10. The main a priori assumptions
  11. The main propositions
  12. Proof of the main Theorem
  13. Eulerian formulation and proof of Proposition \ref{['prodecay']}
  14. The equations in Eulerian coordinates
  15. Strategy for the proof of Proposition \ref{['prodecay']}
  16. ...and 95 more sections

Key Result

Theorem 1.1

Let $h_0(x) = h(0,x)$ be the initial height of the surface $S_0$, and let $\phi_0 (x) = \phi(0,x)$ be the restriction to $S_0$ of the initial velocity potential. Assume that at the initial time one has

Theorems & Definitions (75)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1: Local existence in Lagrangian coordinates WuAG, Wu1
  • Proposition 2.2: Control on the diffeomorphism $k$
  • Proposition 2.3: Energy estimates for the modified Lagrangian variables
  • Proposition 2.4: Transition to Eulerian coordinates
  • Proposition 2.5: Control of dispersive and lower Sobolev norms
  • Proposition 4.1: Poincaré-Shatah Normal Form
  • ...and 65 more